174 HISTORY OP THE THEORY OF NUMBERS. [CHAP, vi
m— r, while r digits precede the period. The condition that the length of the period be the maximum 0(6') is that 10 be a primitive root of bf, whence fc'=pn, since &'^4 or 2pn, p being an odd prime.
P. Bachmann101 used a primitive root g of the prime p and set
to the base g. We get the multiples Q, 2Q, . . . , (p — 1)Q by cyclic permutation of the digits of Q. For p = 7, g = 10, Q « 142857.
J. Kraus102 generalized the last result. When r\jn is converted into a periodic fraction to base g, prime to- n, let aa, . . . , ak be the quotients and TI, . . . , rk the remainders. Then
whence
*\(ai0*-x+ •
In particular, let n be such that it has a primitive root g, and take ri = Then
and if rx is prime to n, the product rxQ has the same digits as Q permuted cyclically and beginning with ax.
H. Brocard103 gave a tentative method of factoring 10n— 1.
J. Mayer104 gave conditions under which the period of z/P to base a, where z and a are relatively prime to P, shall be complete, i. e., corresponding digits of the two halves of the period have the sum a — 1.
Heinrich Bork105 gave an exposition, without use of the theory of numbers, of known results on decimal fractions. There is here first published (pp. 36-41) a table, computed by Friedrich Kessler, showing for each prime p< 100000 the value of g = (p — l)/e, where e is the length of the period for I/ p. The cases in which q — l or 2 were omitted for brevity. He stated that there are many errors in the table to 15000 by Reuschle.40 Cunningham124 listed errata in Kessler 's table.
L. E. Dickson106 proved, without the use of the concept of periodic fractions, that every integer of D digits written to the base N, which is such that its products by D distinct integers have the same D digits in the same cyclic order, is of the form A(ND — 1)/P, where A and P are relatively prime. A number of this form is an integer only when P is prime
101Zeitschrift Math. Phys., 36, 1891, 381-3; Die Elemente der Zahlentheorie, 1892, 95-97.
Alike discussion occurs in 1'interme'diaire des math., 5, 1898, 57-8; 10, 1903, 91-3. l02Zeitschrift Math. Phys., 37, 1892, 190-1. 103E1 Progreso Matematico, 1892, 25-27, 89-93, 114-9. Cf. 1'interme'diaire des math., 2, 1895,
323-4.
104Zeitschrift Math. Phys., 39, 1894, 376-382. 10&Periodische Dezimalbriiche, Progr. 67, Prinz Heinricha-Gymn., Berlin, 1895, 41 pp.